We study the fundamental limits to the expressive power of neural networks. Given two sets $F$, $G$ of real-valued functions, we first prove a general lower bound on how well functions in $F$ can be approximated in $L^p(\mu)$ norm by functions in $G$, for any $p \geq 1$ and any probability measure $\mu$. The lower bound depends on the packing number of $F$, the range of $F$, and the fat-shattering dimension of $G$. We then instantiate this bound to the case where $G$ corresponds to a piecewise-polynomial feedforward neural network, and describe in details the application to two sets $F$: Hölder balls and multivariate monotonic functions. Beside matching (known or new) upper bounds up to log factors, our lower bounds shed some light on the similarities or differences between approximation in $L^p$ norm or in sup norm, solving an open question by DeVore et al. (2021). Our proof strategy differs from the sup norm case and uses a key probability result of Mendelson (2002).

In this article, we present a framework for designing neural networks that remain consistent with the underlying principles of agent-based models. We begin by highlighting the limitations of standard neural differential equations in modeling complex systems, where physical invariants (like energy) are often absent but other constraints (like mass conservation, information locality, bounded rationality) must be enforced. To address this, we introduce Agent-Based-Model informed Neural Networks (ABM-NNs), which leverage restricted graph neural networks and hierarchical decomposition to learn interpretable, structure-preserving dynamics. We validate the framework across three case studies of increasing complexity: (i) a Generalized Lotka-Volterra system, where we recover ground-truth parameters from short trajectories in presence of interventions; (ii) a graph-based SIR contagion model, where our method outperforms state-of-the-art graph learning baselines (GCN, GraphSAGE, Graph Transformer) in out-of-sample forecasting and noise robustness; and (iii) a real-world macroeconomic model of the ten largest economies, where we learn coupled GDP dynamics from empirical data and demonstrate counterfactual analysis for policy interventions.

Drug recommendation is an essential task in machine learning-based clinical decision support systems. However, the risk of drug-drug interactions (DDI) between co-prescribed medications remains a significant challenge. Previous studies have used graph neural networks (GNNs) to represent drug structures. Regardless, their simplified discrete forms cannot fully capture the molecular binding affinity and reactivity. Therefore, we propose Multimodal DDI Prediction with Molecular Electron Localization Function (ELF) Maps (MMM), a novel framework that integrates three-dimensional (3D) quantum-chemical information into drug representation learning. It generates 3D electron density maps using the ELF. To capture both therapeutic relevance and interaction risks, MMM combines ELF-derived features that encode global electronic properties with a bipartite graph encoder that models local substructure interactions. This design enables learning complementary characteristics of drug molecules. We evaluate MMM in the MIMIC-III dataset (250 drugs, 442 substructures), comparing it with several baseline models. In particular, a comparison with the GNN-based SafeDrug model demonstrates statistically significant improvements in the F1-score (p = 0.0387), Jaccard (p = 0.0112), and the DDI rate (p = 0.0386). These results demonstrate the potential of ELF-based 3D representations to enhance prediction accuracy and support safer combinatorial drug prescribing in clinical practice.

Assume that we observe data sampled from pi-VAE model defined according to equation 1, 2 with Poisson noise and parameters =( f, T,) . Assume the following holds: i) The firing rate function f in equation 1 is injective. Because the Bernoulli model is identifiable, the Poisson model is also identifiable. Then we applied two GIN blocks. GIN block, and randomly permuted the input before passing it through each GIN block.